Related papers: Uniform approximation on the sphere by least squar…

The direct and inverse theorems are established for the best approximation in the weighted $L^p$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups. The theorems are stated…

We develop new elements of harmonic analysis on the complex sphere on the basis of which Bernstein's, Jackson's and Kolmogorov's inequalities are established. We apply these results to get order sharp estimates of $m$-term approximations.…

Given a set $P$ of points and a set $U$ of axis-parallel unit squares in the Euclidean plane, a minimum ply cover of $P$ with $U$ is a subset of $U$ that covers $P$ and minimizes the number of squares that share a common intersection,…

Let $\zeta$ be a real transcendental number. We introduce a new method to find upper bounds for the classical exponent $\widehat{w}_{n}(\zeta)$ concerning uniform polynomial approximation. Our method is based on the parametric geometry of…

We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical…

We obtained order estimations for the best uniform approximations by trigonometric polynomials and approximations by Fourier sums of classes of $2\pi$-periodic continuous functions, which $(\psi,\beta)$-derivatives $f_{\beta}^{\psi}$ belong…

We investigate projection constants for spaces of bihomogeneous harmonic and bihomogeneous polynomials on the unit sphere in finite-dimensional complex Hilbert spaces. Using averaging techniques, we demonstrate that the minimal norm…

There were established the exact-order estimations of the best uniform approximations by{\psi} the trigonometrical polynoms on the $C^{\psi}_{\beta,p}$ classes of $2\pi$-periodic continuous functions $f$, which are defined by the…

Given a measure $\mu$ on the unit sphere $\partial\mathbb{B}^d$ in $\mathbb{C}^d$ with Lebesgue decomposition ${\rm d} \mu = w \, {\rm d} \sigma + {\rm d} \mu_s$, with respect to the rotation-invariant Lebesgue measure $\sigma$ on $\partial…

A fully implementable filtered polynomial approximation on spherical shells is considered. The method proposed is a quadrature-based version of a filtered polynomial approximation. The radial direction and the angular direction of the…

We consider the approximation of a continuous function, defined on a compact set of the $d$-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best…

We analyze the accuracy of the discrete least-squares approximation of a function $u$ in multivariate polynomial spaces $\mathbb{P}_\Lambda:={\rm span} \{y\mapsto y^\nu \,: \, \nu\in \Lambda\}$ with $\Lambda\subset \mathbb{N}_0^d$ over the…

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial $f$ over the boolean hypercube $\mathbb{B}^{n}=\{0,1\}^n$. This hierarchy provides for each integer $r \in \mathbb{N}$ a lower bound…

Let $\mathscr{C}_\mathbb{Z}([0,1])$ be the metric space of real-valued continuous functions on $[0,1]$ with integer values at $0$ and $1$, equipped with the uniform (supremum) metric $d_\infty$. It is a classical theorem in approximation…

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is…

We study comparison of Lp norms of polynomials on the sphere with respect to doubling measures. From our description it follows an uncertainty principle for square integrable functions on the sphere. We consider also weighted uniform…

We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the…

We construct explicit easily implementable polynomial approximations of sufficiently high accuracy for locally constant functions on the union of disjoint segments. This problem has important applications in several areas of numerical…

We provide a proof of effective uniformization for nearly round 2-spheres, utilizing an identity related to the third-order differential of the conformal factor. This identity is connected to the geometry of the embedded spacelike surface…

We find the precise range of $(p,q)$ for which local averages along graphs of a class of two-variable polynomials in $\mathbb{R}^3$ are of restricted weak type $(p,q)$, given the hypersurfaces have Euclidean surface measure. We derive these…